steinitz representations lászló lovász microsoft research one microsoft way, redmond, wa 98052...
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Steinitz Representations
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
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Steinitz 1922
Every 3-connected planar graphis the skeleton of a convex 3-polytope.
3-connected planar graph
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Coin representation
Every planar graph can be represented by touching circles
Koebe (1936)
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Polyhedral version
Andre’ev
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
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From polyhedra to circles
horizon
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From polyhedra to representation of the dual
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Rubber bands and planarity
G: 3-connected planar graph
outer face fixed toconvex polygon
edges replaced byrubber bands
2( )i jij E
u u
EEnergy:
Equilibrium:( )
1i j
j N ii
u ud
Tutte (1963)
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G 3-connected planar
rubber band embedding is planar
Tutte
(Easily) polynomial time computable
Lifts to Steinitz representation
Maxwell-Cremona
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G=(V,E): connected graph
M=(Mij): symmetric VxV matrix
Mii arbitraryMij
<0, if ijE
0, if ,ij E i j
weighted adjacency matrix of GG-matrix
: eigenvalues of M1 2 1... ...k n 0
WLOG
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G planar, M G-matrix
corank of M is at most 3.
Colin de VerdièreVan der Holst
G has a K4 or K2,3 minor
G-matrix M such that
corank of M is 3.
Colin de Verdière
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Proof.
(a) True for K4 and K2,3.
(b) True for subdivisions of K4 and K2,3.
(c) True for graphs containing subdivisions of K4 and K2,3.
Induction needs stronger assumption!
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rk( ) rk( )A M
0 forijA ij E
transversal intersection
M
VxV symmetric matrices
Strong Arnold property
( )ijX X symmetric,
X=00ijX ij E i j for and
0,MX
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Representation of G in 3
Nullspace representation
0ij jj
M u
basis of nullspace of M1 2 3 :x x x
11 21 31
12 22
1
232
12 22 3n n
x x x
x x x
ux
u
u
x x
1( )( ) 0i ij j j jj
c M c c u scaling M scaling the ui
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Van der Holst’s Lemma
connected
like convex polytopes?
or…
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Van der Holst’s Lemma, restated
Let Mx=0. Then
sup ( ), sup ( )x x
are connected, unless…
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G 3-connected planar
nullspace representationcan be scaled to convex polytope
G 3-connected planar
nullspace representation,scaled to unit vectors,gives embedding in S2
L-Schrijver
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planar embedding nullspace representation
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Stresses of tensegrity frameworks
bars
struts
cables x y( )ijM x y
( ) 0ij j ij
M x x Equilibrium:
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Cables
Braced polyhedra
Bars
0
0
0 ( , , )ij
ii i ijj V
i j V ij E
M
M
M M
0ij jj V
M u
stress-matrix
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There is no non-zero stress on the edges of a convex polytope
Cauchy
Every braced polytopehas a nowhere zero stress (canonically)
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( )uvMp q u v
( ) ( )
( ) 0edge
of u
uv uvv N u v N u pq
F
u M v M u v p q
( )uv
v N uuuM v uM
q
p
uFu v
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The stress matrix of anowhere 0 stress on a braced polytope
has exactly one negative eigenvalue.
The stress matrix of aany stress on a braced polytope
has at most one negative eigenvalue.
(conjectured by Connelly)
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Proof: Given a 3-connected planar G, true for
(a) for some Steinitz representation and the canonical stress;
(b) every Steinitz representation and the canonical stress;
(c) every Steinitz representation and every stress;
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Problems
1. Find direct proof that the canonical stress matrix has only 1 negative eigenvalue
2. Directed analog of Steinitz Theorem recently proved by Klee and Mihalisin. Connection with eigensubspaces of non-symmetric matrices?
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Let .
Let span a components;
let span b components.
Then , unless…
3. Other eigenvalues?
sup ( )x
kMx x
sup ( )x
a b k
From another eigenvalue of the dodecahedron,we get the great star dodecahedron.
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4. 4-dimensional analogue?
(Colin de Verdière number): maximumcorank of a G-matrix with the Strong Arnoldproperty
( )G
( ) 3G G planar
( ) 4G G is linklessly embedable in 3-space
LL-Schrijver
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Linklessly embeddable graphs
homological, homotopical,…equivalent
embeddable in 3 without linked cycles
Apex graph
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Basic facts about linklessly embeddable graphs
Closed under:
- subdivision
- minor
- Δ-Y and Y- Δ transformations
G linklessly embeddable
G has no minor in the “Petersen family”
Robertson – Seymour - Thomas
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The Petersen family
(graphs arising from K6 by Δ-Y and Y- Δ)
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Can it be decided in P whethera given embedding is linkless?
Can we construct in P a linkless embedding?
Is there an embedding that canbe certified to be linkless?
Given a linklessly embedable graph…
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